We discuss family unification in grand unified theory (GUT) based on an GUT gauge group broken to its subgroups, including a special subgroup. In the GUT on the six-dimensional (6D) orbifold space , three generations of the 4D Standard Model Weyl fermions can be embedded into 6D bulk Weyl fermions in an second-rank anti-symmetric tensor representation. 6D and 4D gauge anomalies can be canceled out by considering proper matter content without 4D exotic chiral fermions at low energies.
B40B41B42B431. Introduction
The existence of three chiral generations of quarks and leptons is one of the most mysterious facts in particle physics. In addition, their hierarchical mass structures generated by the Higgs mechanism via their corresponding Yukawa couplings strongly suggest the existence of a hidden structure of nature. There have been many attempts to understand the origin of chiral generations and/or their hierarchical mass structures by considering, e.g., so-called horizontal symmetry (or family symmetry) in four-dimensional (4D) theories [1–7], geometrical structures in higher-dimensional theories [8–10], and string theories [11–13].
As is well known, quarks and leptons for each generation in the Standard Model (SM) can be unified into one multiplet (or two multiplets) in grand unified theories (GUTs) [14]. There are many GUTs in the 4D framework [14–19] and higher-dimensional space frameworks [20–32] (for a review, see Refs. [33,34]).
There have been some attempts to unify GUT and family groups into a larger GUT group in 4D and higher-dimensional theories [35–42]. However, such attempts are based on GUT groups and their limited subgroups, so-called regular subgroups, e.g., . There are other subgroups called special subgroups (or non-regular subgroups), e.g., , , , and . (For Lie groups and their subgroups, see, e.g., Refs. [33,34,43–49].)
Recently, new-type GUTs called special GUTs based on GUT groups and and their special subgroup have been proposed in Refs. [50,51]. The main results of and special GUTs are summarized as follows. In an special GUT based on its GUT group broken to its special subgroup , a 4D Weyl fermion can be identified with one generation of quarks and leptons; 4D gauge anomaly cancellation does not work in the 4D framework, while it works in the 6D framework without any exotic chiral fermions once we take into account symmetry breaking effects [50]. Almost the same results are obtained in an special GUT [51].
In a special GUT framework, family unification can be considered by using GUT groups and their “regular-type” and “product-type” subgroups; an example of the former is ; an example of the latter is , where contains an ordinary GUT gauge group and is a family gauge group. (Their branching rules of , , etc. can be calculated, e.g., by using the projection matrix method shown in Refs. [34,45,46].)
First, for a regular-type case, an example of GUT gauge groups and their subgroup pair is . This is a simple extension of . The branching rule of for the defining representation is
In this case, e.g., an second-rank anti-symmetric tensor representation contains three generations of quarks and leptons. Its branching rule is given by
where an representation is complex while an representation is real. A 4D Weyl fermion in an representation is vectorlike, so when we take into account symmetry breaking effects for to , only three 4D Weyl fermions remain chiral. Also, an second-rank symmetric tensor representation contains an representation. Note that the contains unwilling complex representations. The adjoint representation contains not only an but also its conjugate representation . (In Ref. [52], R. M. Fonseca has also pointed out that a 4D Weyl fermion in an representation contains the SM fermions plus vectorlike particles only, which was found by using the Susyno program [48].)
Next, for a product-type case, an example of GUT gauge groups and their subgroup pair is . The branching rule for the defining representation is
The 4D Weyl fermion in the defining representation can be identified with three chiral generations of quarks and leptons. To the best of my knowledge, there is no way to construct an gauge theory that contains only three chiral generations without any gauge anomalies, at least for a 4D, 5D, or 6D framework. We will not discuss this possibility in this letter.
In this letter we discuss a 6D special GUT on , whose GUT group includes an GUT group and an family group. The main purpose of this paper is to show that three generations of the 4D SM Weyl fermions can be embedded into 6D bulk Weyl fermions in an second-rank anti-symmetric tensor representation. 6D and 4D gauge anomalies can be canceled out by considering proper matter content without 4D exotic chiral fermions at low energies.
This letter is organized as follows. In Sect. 2, before we discuss a special GUT based on an gauge group, we discuss basic properties of and its subgroups mainly by using the technique in Ref. [34]. In Sect. 3 we construct a 6D special GUT on . Section 4 is devoted to a summary and discussion.
2. Basics for <inline-formula><tex-math notation="LaTeX" id="ImEquation72"><![CDATA[$SU(19)$]]></tex-math></inline-formula> and its subgroups
First, we check how to embed three generations of the SM Weyl fermions into 4D Weyl fermion in an second-rank anti-symmetric tensor representation . For regular and special embeddings , an second-rank anti-symmetric tensor representation is decomposed into an second-rank anti-symmetric tensor representation , defining representations , and bi-fundamental representations given in Eq. (1.2). Further, as is well known, the spinor representation is decomposed into representations:
That is, three generations of the SM Weyl fermions are embedded into a 4D Weyl fermion. In addition, an complex representation is identified with an real representation . A 4D Weyl fermion is chiral, while a 4D Weyl fermion is vectorlike. When is broken to , is broken to nothing. A 4D Weyl fermion in the is chiral, while three 4D Weyl fermions are vectorlike. Thus, once is broken to , a 4D Weyl fermion is decomposed into three 4D Weyl fermions and vectorlike fermions.
Next, we consider a symmetry breaking pattern from to . One way of achieving this is to use orbifold symmetry breaking boundary conditions (BCs) and several GUT-breaking Higgses. One example is to choose orbifold BCs breaking to and to introduce , , scalar fields, where we assume their proper components acquire non-vanishing VEVs (see Fig. 1). First, the following orbifold BC for the defining representation breaks to :
where stands for a projection matrix defined in Eq. (3.5). The non-vanishing VEV of the scalar field is responsible for breaking to , where its branching rule of is given by
A symmetry breaking pattern of to . BCs stands for an orbifold boundary condition. represents a scalar field in a representation of . We assume that the appropriate component of each develops its non-vanishing VEV .
An contains a singlet under its special subgroup. Its non-vanishing VEV can break to its special subgroup [50,51], where their decompositions are given in Ref. [34] by
The VEV of an scalar breaks to or to , where its breaking rule is given in Eq. (1.1). If the three VEVs of the proper components of scalars can break to , the VEV of the scalar further reduces to , where its branching rule of is given by
where the is decomposed into , and the is decomposed into . (For further information, see, e.g., Ref. [34].)
3. <inline-formula><tex-math notation="LaTeX" id="ImEquation164"><![CDATA[$SU(19)$]]></tex-math></inline-formula> special grand unification
As in Refs. [50,51], we consider an special GUT on 6D orbifold spacetime with the Randall–Sundrum (RS) type metric [31,32,53] given by
where is the coordinate of RS warped space, is the coordinate of , , , for , and . There are four fixed points on at , , , and . For each fixed point, the parity reflection is described by
where , and . The fifth and sixth dimensional translations and satisfy and , respectively.
We consider the matter content in the special GUT that consists of a 6D bulk gauge boson : 6D positive Weyl fermions with the orbifold BCs and and , and 6D negative Weyl fermions with and and , where stands for parity assignment for each 6D fermion; 5D , , and brane scalar bosons at , , , ; a 4D Weyl fermion, four 4D Weyl fermions, 64 4D Weyl fermions, four 4D Weyl fermions at the fixed point , , , . The matter content of the special GUT is summarized in Table 1. Note that the 5D brane scalars are responsible for achieving the appropriate symmetry and the 4D brane fermions are necessary to realize 4D gauge anomaly cancellation. Only their conditions do not uniquely determine the matter content, so one may choose another matter content; e.g., one may introduce 6D bulk scalars instead of 5D brane scalars. We will see the roles of the bulk and brane fields in the following.
The matter content in the special GUT on . The representations of and 6D, 5D, 4D Lorentz group, the orbifold BCs of 6D bulk fields and 5D brane fields, and the spacetime location of 5D and 4D fields are shown. Orbifold BCs stand for parity assignment for 6D fields and for 5D fields. The orbifold BCs of the 6D gauge field are given in Eqs. (3.3) and (3.4). ; ; ; .
6D bulk field
Orbifold BC
5D brane field
1
1
1
1
Orbifold BC
Spacetime
4D field
Spacetime
First, a 6D bulk gauge boson is decomposed into a 4D gauge field and fifth- and sixth-dimensional gauge fields and . The orbifold BCs of the 6D gauge field are given by
where is a projection matrix satisfying . We consider the orbifold BCs and preserving symmetry, while the orbifold BCs and reduce to its regular subgroup . We take as
In this case, the 4D gauge field have Neumann BCs at the fixed points and , while the fifth- and sixth-dimensional gauge fields and have Dirichlet BCs because of the negative sign in Eq. (3.3). On the other hand, since symmetry is broken to at the fixed points and , by using the branching rules of the adjoint representation given in Eq. (2.5) as well as Eqs. (3.3) and (3.4), the and components of the 4D gauge field have Neumann and Dirichlet BCs at the fixed points and , respectively; the and components of the fifth- and sixth-dimensional gauge fields and have Dirichlet and Neumann BCs, respectively. Thus, since the components of the 4D gauge field have four Neumann BCs at the four fixed points , they have zero modes corresponding to 4D , , and gauge fields; since the other components of and any component of and have four Dirichlet BCs or two Neumann and two Dirichlet BCs at the four fixed points, they do not have zero modes. The orbifold BCs reduce to .
To achieve the SM gauge symmetry at low energies, we consider the symmetry breaking sector via spontaneous symmetry breaking. We introduce 5D , , and brane scalar fields, , , , and on the 5D brane (). Their orbifold BCs are given by
where , stands for , , and , is a positive or negative sign, and is a projection matrix. We take . The branching rules of for , , and are given in Eqs. (2.3), (2.5), and (1.1), respectively. For , the components have zero modes; for , the components have zero modes; for , the components have zero modes; and for , the components have zero modes. We assume that a scalar field is responsible for breaking to ; two scalar fields are responsible for breaking to ; the non-vanishing VEV of the scalar field breaks to ; the non-vanishing VEV of breaks to .
The SM Weyl fermions are identified with zero modes of a 6D Weyl bulk fermion. The orbifold BCs of 6D positive or negative Weyl bulk fermions can be written by
where the subscript of stands for 6D chirality, is a positive or negative sign, , the 6D gamma matrices satisfy (), , , , and is a projection matrix. (The same notation is used in Refs. [31,32].) In our notation, a 6D Dirac fermion and 6D positive and negative Weyl fermions () can be expressed by using 4D left- and right-handed Weyl fermions (), where the subscripts and stand for 4D and 6D chiralities, respectively:
Note that to see the relation between and , we express the orbifold BCs of by using a matrix form, the same as that of the gauge field in Eq. (3.3). We can write the component of as , where the s are the component fields of expanded by s, and . In this notation, the orbifold BCs of the 6D anti-symmetric tensor fermion can be expressed by using the projection matrix given in Eq. (3.4) instead of :
where , and denotes the element of the projection matrix given in Eq. (3.4). (The projection matrix of any tensor product representation can be expressed by the projection matrix of the defining representation .)
Here, we check zero modes of, e.g., a 6D positive Weyl fermion with orbifold BCs . At fixed points and , the 4D left-handed Weyl fermion components have Neumann BCs, while the 4D right-handed Weyl fermion components have Dirichlet BCs. At fixed points and , the 4D and left-handed Weyl fermion components have Neumann and Dirichlet BCs, respectively, while the 4D and right-handed Weyl fermion components have Dirichlet and Neumann BCs, respectively. In this case, only the 4D left-handed Weyl fermion has zero modes. Also, for , the 4D right-handed Weyl fermion has zero modes. They are vectorlike once we take into account the symmetry breaking effects of to . Also, for and , there is no zero mode. The parity assignments of are summarized in Table 2.
The parity assignments of the 4D left- and right-handed Weyl fermion components of the 6D positive and negative Weyl fermions , , , and given in Eq. (3.6).
Left
Right
Left
Right
Left
Right
Left
Right
Here, we check the contribution to 6D bulk and 4D brane anomalies from the above 6D Weyl fermion sets. The fermion set does not contribute to 6D gauge anomalies because of the same number of 6D positive and negative Weyl fermions. We need to check 4D gauge anomaly cancellation at four fixed points by using the 4D anomaly coefficients listed in Ref. [34]. From Table 2, at three fixed points there are two 4D left- and right-handed Weyl fermions in , , and of from the 6D positive and negative Weyl fermions , , , and . The vectorlike matter sets do not produce any 4D gauge anomalies. At the other fixed point , there can be 4D pure , pure , pure , mixed , mixed , and mixed anomalies. In fact, the 6D positive and negative Weyl fermions generate 4D pure , pure , pure , mixed , mixed , and mixed anomalies. We focus on how to cancel the 4D anomalies at the fixed point below.
To achieve 4D gauge anomaly cancellation at the fixed point , we need to introduce 4D Weyl fermions in appropriate representations of . First, we consider the pure gauge anomaly cancellation. The 4D gauge anomaly of 12 4D left-handed Weyl fermions and four 4D right-handed Weyl fermions is canceled out by the anomaly of three 4D left-handed Weyl fermions. From Table 1, there are one 4D left-handed Weyl fermion and four 4D left-handed Weyl fermions in the model. The gauge anomaly of is canceled out by one of, e.g., . The other three 4D left-handed Weyl fermions contribute the gauge anomaly on the fixed point . Thus, the 4D brane fermions and cancel the 4D gauge anomaly from the bulk fermions. Second, for the pure gauge anomaly, the 4D gauge anomaly of 64 4D left-handed Weyl fermions and four 4D right-handed Weyl fermions is canceled out by the anomaly of 68 4D left-handed Weyl fermions. Third, the 4D pure , mixed , mixed , and mixed anomalies are canceled out if the matter content is vectorlike from the view of the gauge theory. The matter content shown in Table 1 satisfies all the above requirements, so any 6D and 4D gauge anomalies at the fixed points are canceled out.
4. Summary and discussion
In this letter we have pointed out that in a special GUT framework, family unification may be achieved by using GUT groups and their “regular-type” and “product-type” subgroups, such as and , respectively.
For a “regular-type” subgroup , we have constructed an special GUT by using a special breaking to . In this framework, the zero modes of a 6D Weyl fermion can be identified with three generations of quarks and leptons; the 6D gauge anomaly on the bulk and the 4D or gauge anomalies at each fixed point can be canceled out; as in the special GUT [50], exotic chiral fermions do not exist due to a special feature of the complex representation once we take into account the symmetry breaking of to .
To cancel 4D pure , pure , pure , and mixed anomalies on a fixed point, we introduced a lot of 4D Weyl fermions. For the mixed anomalies, one may rely on the Green–Schwarz anomaly cancellation mechanism [54] for the 4D version [55,56]. It may be achieved by introducing a pseudo-scalar field that transforms non-linearly under the anomalous symmetry. In this case, the number of 4D Weyl fermions can be drastically reduced.
We comment on the SM fermion masses in the special GUT. Since three generations of the SM fermions are unified into a 6D Weyl fermion, the masses of all quarks and leptons are degenerate without breaking effects. We assumed that since the non-vanishing VEVs of 5D brane scalars break the symmetry, there is no reason to expect the unified masses of first, second, and third generations of up-type and down-type quarks, charged leptons, and neutrinos, respectively; in addition, since the non-vanishing VEVs of 5D brane scalars , , and break to , there is no reason to expect the degenerate mass of quarks and leptons for each generation. As discussed in, e.g., Refs. [31,32], on the UV brane , we can introduce -invariant brane interaction terms among the 6D bulk fermions and the 5D brane scalars because tensor products, e.g., , , , etc. contain a singlet. The and components of the bulk fermions can be mixed via the VEVs of the 5D brane scalars once their corresponding brane interaction terms or effective brane mass terms are generated, where contains , , , , and of . We expect that the effective mass terms divide the degenerate mass for each generation into up-type quark, down-type quark, charged lepton, and neutrino masses, where some VEVs or coupling constants must be hierarchical to realize mass hierarchies for up-type and down-type quarks and charged leptons. To realize tiny neutrino masses, it seems to be better to introduce 5D symplectic Majorana fermions [57] on the UV brane . brane interaction terms among the 6D bulk fermions, the 5D brane scalars, and the 5D symplectic Majorana fermions lead to tiny neutrino masses via a seesaw mechanism as discussed in Refs. [31,58]. The above brane interaction terms are essential to realize not only quark and lepton masses but also their mixing matrices, i.e., the Cabibbo–Kobayashi–Maskawa [59,60] and Maki–Nakagawa–Sakata [61] matrices. Since we can introduce a lot of 5D brane interaction terms in the special GUT, the model seems to realize the SM fermion masses, but seems to give us no prediction about quark and lepton masses and mixings. We will leave the detailed analysis for future studies.
We have discussed how to embed three chiral generations of quarks and leptons in a triplet (a finite-dimensional representation) of a non-Abelian compact group . Another direction for unifying generations may be considered by using non-Abelian non-compact groups (e.g., ) and their infinite-dimensional representation [5,62–68].
Acknowledgements
The author would like to thank Kenji Nishiwaki for critical reading of the manuscript and many valuable comments.
Funding
Open Access funding: SCOAP.
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